Law of Cosines Calculator

Law of Cosines
Side a =
Side b =
Side c =
Angle Units in:
Length Units in:
Results Significant Figures
a =
b =
c =

A =
B =
C =

P =
s =
K =
r =
R =

This Calculation Equation & Triangle

\( \large A = \cos^{-1} \left[ \frac {b^2+c^2-a^2} {2bc} \right] \)

Triangle Diagram with Angles A, B and C and sides opposite those angles a, b and c respectively
A = angle A
B = angle B
C = angle C
a = side a
b = side b
c = side c
P = perimeter
s = semi-perimeter
K = area
r = radius of inscribed circle
R = radius of circumscribed circle

Law of Cosines Calculator

Uses the law of cosines to calculate unknown angles or sides of a triangle.  In order to calculate the unknown values you must know and enter 3 known values. 

To calculate any angle, A, B or C, enter 3 side lengths a, b and c.  This is the same calculation as Side-Side-Side (SSS) Theorem.  To calculate a side, say a, enter the opposite angle A and the to other adjacent sides b and c.  Using different forms of the law of cosines we can calculate all of the other unknown angles or sides. This is the same calculation as Side-Angle-Side (SAS) Theorem.

Base Triangle ABC.

Law of Cosines

If a, b and c are the lengths of the legs of a triangle opposite to the angles A, B and C respectively; then the law of cosines states:

\[ a^2 = b^2 + c^2 - 2bc \cos A \] \[ b^2 = a^2 + c^2 - 2ac \cos B \] \[ c^2 = a^2 + b^2 - 2ab \cos C \]

Law of Cosines solving for sides a, b, and c

\[ a = \sqrt {b^2 + c^2 - 2bc \cos A } \] \[ b = \sqrt {a^2 + c^2 - 2ac \cos B } \] \[ c = \sqrt {a^2 + b^2 - 2ab \cos C } \]

Law of Cosines solving for angles A, B, and C

\[ A = \cos^{-1} \left[ \frac {b^2+c^2-a^2} {2bc} \right] \] \[ B = \cos^{-1} \left[ \frac {a^2+c^2-b^2} {2ac} \right] \] \[ C = \cos^{-1} \left[ \frac {a^2+b^2-c^2} {2ab} \right] \]

Triangle Characteristics

Triangle perimeter, P = a + b + c

Triangle semi-perimeter, s = 0.5 * (a + b + c)

Triangle area, K = √[ s*(s-a)*(s-b)*(s-c)]

Radius of inscribed circle in the triangle, r = √[ (s-a)*(s-b)*(s-c) / s ]

Radius of circumscribed circle around triangle, R = (abc) / (4K)

References/ Further Reading

Weisstein, Eric W. "Law of Cosines" From MathWorld-- A Wolfram Web Resource.

Zwillinger, Daniel (Editor-in-Chief). CRC Standard Mathematical Tables and Formulae, 31st Edition New York, NY: CRC Press, p. 512, 2003.


Cite this content, page or calculator as:

Furey, Edward "Law of Cosines Calculator" From - Online Calculator Resource.